Lagrangian for Real Systems Instead of Entropy for Ideal Isolated Systems

Abstract

The Second Law of Thermodynamics states that entropy S increases in a spontaneous process in an ideal isothermal and isolated system. Real systems are influenced by external forces and fields, including the temperature field. In this case, only entropy is not enough, and we suggest using a new function, $L_s$, which is analogous to the Lagrangian in classical mechanics. It includes total potential energy but instead of mechanical kinetic energy, $L_s$ includes the product ST, and the system always evolves towards increasing this modified Lagrangian. It reaches an equilibrium when total potential force is balanced by both entropic and thermal forces. All forces have the same units, Newton/mol, and may be added or subtracted. For condensed systems with friction forces, it is a molecular transport velocity, and not acceleration, which is proportional to the acting force. Our approach has several advantages compared to Onsager’s non-equilibrium thermodynamics with its thermodynamic forces, which may have different units, including 1/T for energy transport. For isolated systems, the description is reduced to Second Law and Clausius inequality. It easily explains diffusion, Dufour effect, and Soret thermodiffusion. The combination of electric, thermal, and entropic forces explains thermoelectric phenomena, including Peltier–Seebeck and Thomson (Lord Kelvin) effects. Gravitational and entropic forces together inside a black hole may lead to a steady state or the black hole evaporation. They are also involved in and influenced by solar atmospheric processes.

1. Introduction

In 2024, it has been two hundred years since Sadi Carnot published his “Reflections on the Motive Power of Heat”. Since that time Clausius, Kelvin, Maxwell, Boltzmann, Gibbs, and many other physicists and chemists have been working on entropy-based formulation and development of the Second Law of Thermodynamics for isolated systems. There are many ways to formulate the Second Law of Thermodynamics. For example, one of the axiomatic statements is: “There exists for every thermodynamic system in equilibrium an extensive scalar property called the entropy $S$, such that in an infinitesimal reversible change of state of the system, $dS=dQ/T$, where $T$ is the absolute temperature and $dQ$ is the amount of heat received by the system. The entropy of a thermally insulated system cannot decrease and is constant only if all processes are reversible” [1]. A more accurate term would be an isolated system.

Here, we should remind that equilibrium Gibbs thermodynamics has three major laws, and it answers the question “How much?” but not “How fast?”. Newtonian mechanics also is based on three laws. It answers the question “How fast?” Indeed, Newton’s Second Law is $a=F/m$, where acceleration $a$ in an ideal system is proportional to the acting force $F$.

Presented here approach, which we call physicochemical mechanics, for condensed fluids with friction leads to the motion equation, where it is velocity and not acceleration, which is proportional to total of acting forces. All conservative molecular forces have the same units, newton/mol, and may be added, as in Newtonian mechanics. To calculate any transport coefficient, all we need is mobility and conjugated molar properties, such as molar charge, molar volume, etc. This is a big advantage in comparison to Onsager’s non-equilibrium linear thermodynamics, where half of transport coefficients from the symmetric matrix must be determined in experiments, and so-called thermodynamic forces must be found based on the rates of entropy production.

Note that traditional equilibrium thermodynamic potentials like Gibbs or Helmholtz free energy were developed for isothermal conditions and should not be used for systems with the temperature gradients [2]. The purpose of this paper is, based on Newtonian forces, to consider a more realistic case of molecular transport in systems with friction, which are not in equilibrium and are influenced both by external fields, including temperature gradients, and internal entropy gradients.

2. Methods

Figure 1 shows an isolated system (a) and a non-isolated system (b). If there are several forces, we should use ${\displaystyle \sum F_j}$. Of course, the isolated system is idealization valid for a vacuum or a planetary system. In real, especially condensed, systems, we should add a friction force, which is proportional to velocity, but has an opposite direction.

Very soon it becomes equal to the total of all applied forces. As a result, they balance each other, and the transport process reaches a steady state when the acceleration vanishes. After substitution, it is the velocity and not acceleration which becomes proportional to the total of all acting forces. In hydrodynamics, this is known as Stokes’ equation. Instead of coefficient $1/m$, the proportionality coefficient is $1/\zeta$, where $\zeta$ is proportional to the solution friction and depends on the particle shape [3]. To describe Brownian motion and diffusion of molecules, Einstein suggested molecular mobility $U=1/\zeta$, which is velocity per unit of acting molar force with units (cm/s)/(newton/mol) [4]. For transport of each component, we usually need the flux J_i (mol/s per cm²), which is proportional to the product of velocity by concentration, i.e., $J_i=U_i{c}_i{\displaystyle \sum _j{F}_{ji}}$. Without concentration we have an expression for velocity, equivalent to equations of motion in mechanics. If we deal with an ionic flux, the major driving force is the negative gradient of electric potential, multiplied by molar charge $F_{ei}=-z_{i}F{\displaystyle \frac{d{\psi }_e}{dx}}$, leading to Faraday’s law $J_i=-U_i{c}_i{z}_{i}F{\displaystyle \frac{d\psi }{dx}}$. Here, F is the Faraday constant and $z_i$ is an elementary ion charge. The product of J_i and a molar charge gives electric current and leads to Ohm’s law with electric conductivity $U_i{c}_i{(z_{i}F)}^2$.

Physicochemical mechanics [5] allow systematic derivation of all major transport laws and their equilibrium relations. The mobility coefficient is inversely proportional to solution viscosity and may be determined, for example, from a diffusion experiment. Molar properties usually are known, thus giving us all molar transport coefficients. This is a fundamental advantage in comparison to Onsager’s theory, where the matrix of transport coefficients is symmetrical, but we still must conduct experiments to measure half of the transport coefficients. When two different forces become equal but have opposite direction, the process stops, which allows derivation of a bunch of equilibrium laws, such as Nernst’s law for transmembrane electric potential, which is impossible to derive in Onsager’s theory because of different units for different thermodynamic forces.

We have previously described the rates of transport processes influenced by external fields, and also their mutual equilibrium. For seven major driving factors, we suggested a Table with more than 7 × 7 transport laws and their equilibrium relations [5]. The typical value of these driving factors decreases from the left to the right column, and from the upper to the lower horizontal line. The Table may grow further if we add new driving factors, including mechanical deformations of solids or even optical trapping of single molecules.

It is known that, based on his Periodic Table of elements, Mendeleev predicted properties of four yet unknown elements [6]. We predicted one new phenomenon, which we called magneto tension [5]. In this case deformation of the liquid surface by a magnetic field is observed. Later, it was confirmed in experiments, and now it is called the Moses effect [7], drawing upon the description in the Old Testament.

3. Results

3.1. General Equation

While describing transport for a chemical component i in isothermal and not isolated systems, we previously suggested using a new general physicochemical potential, ${\mu }_{gi}$, where ${\mu }_{gi}={\mu }_i+{\displaystyle \sum _{j-1}{\chi }_{j-1}}{\psi }_{j-1}$. Here, ${\mu }_i={\mu }_{0i}+RT\ln c_i$ is the traditional chemical potential of this component. ${\psi }_{j-1}(x)$ is the local field potential and ${\chi }_{j-1}$ is the conjugated molar property. For example, for electric field potential ${\psi }_e$ the molar charge is ${\mathrm{Q}}_i$ and ${\mu }_{ei}={\mu }_i+{\mathrm{Q}}_i{\psi }_e$ is known as an electrochemical potential. The negative gradient $-d{\mu }_{gi}/dx$ gives the total local molar force (newton/mol) leading to transport of i. This potential-based conservative force does not depend on velocity. Without external potential fields or for isolated systems it is based on traditional chemical potential, which is isobaric-isothermal Gibbs energy $G=E-ST+VP$ per mol of respective component. In isothermal conditions without chemical reactions, internal energy $E$ is constant. $VP$ historically comes from an equation for an ideal gas, and now it may be included in the total of all other driving factors.

As long as transport leads to energy dissipation into heat, if local temperature also changes, instead of gradient of chemical potential $-d{\mu }_i/dx$, we suggest using:

$${\displaystyle \frac{d{q}_i}{dx}}={\displaystyle \frac{d(s_{i}T)}{dx}}={\displaystyle \frac{Td{s}_i}{dx}}+{\displaystyle \frac{s_{i}dT}{dx}}$$

(1)

Here, $q_i$ is the local molar heat (not charge!), carried by i-th component, and $s_i$ is the local molar entropy of this component. For molar entropy we use lower case letter s and for system entropy (the extensive function) we use the upper-case letter S. Both terms ${\displaystyle \frac{Td{s}_i}{dx}}$ and ${\displaystyle \frac{s_{i}dT}{dx}}$ have units of molar force and may be called the entropic and thermal forces. They have positive signs, reflecting, for example, that at constant temperature the process leads to an increase of entropy (the Second Law). Both terms reminding magnetic effects are not conservative and depend on rates of molecular Brownian motion.

Many transport processes are conducted in isothermal conditions. If there is a temperature gradient in the sample, we must multiply local temperature by molar entropy and then take a space derivative of the product with the plus sign. This obvious difference is because a system spontaneously evolves towards decrease of potential energy, but towards the increase of entropy. Further, usual molar parameters, like molar volume, stay constant during transport, but molar entropy is not. It is temperature-dependent, so we have a function of two variables and two new forces: $+T\partial s_i/\partial x$ and $+s_i\partial T/\partial x$. In combination with usual conservative forces, which are the gradients of related components of potential energy with the minus sign, this leads to simple explanations of thermodiffusion and thermoelectric phenomena, and, for example, explains why sometimes a species moves to cold, and after slight changes in concentration or temperature, it moves to the hot area. This was not described before.

Here it makes sense to recall the well-known phrase by J.W. Gibbs: “If we wish to find in rational mechanics a priory foundation for the principles of thermodynamics, we must seek mechanical definition of temperature and entropy” [8]. Each conservative potential force $F_{ji}$ is counteracting entropic and thermal forces. In the steady state the total of these forces is balanced by friction forces and the directed mass transport reaches a constant velocity. At equilibrium, the external force minus thermal and entropic forces is zero, no directed friction counterforces are formed, and the total flux vanishes. Further, without external forces, local temperature does not depend on x, as suggested by Clausius $d{s}_i=d{q}_i/T$. For the whole universe, this imaginary state is known as the heat death of the universe.

3.2. Clausius Inequality, Fokker–Planck–Smoluchowski, Nernst, and Van’t Hoff Laws

In the presence of external forces, we have a general equation:

$$J_i=U_i{c}_i({\displaystyle \frac{s_{i}dT}{dx}}+{\displaystyle \frac{Td{s}_i}{dx}}+{\displaystyle \sum _j{F}_{ji}})$$

(2)

To understand this equation better, as usual, we will start with an ideal gas. It is known that at equilibrium molar entropy of an ideal gas per unit volume is $\mathrm{s}\hspace{0.17em}{=\mathrm{s}}_o-R\ln c+C_{\mathrm{v}}\ln T$ [2], where $C_{\mathrm{v}}$ is the molar heat capacity at constant volume. This expression should also be valid for local entropy without equilibrium. Both ${\mu }_i$ and $s_i$ include the term Rlnc but with opposite signs. Without ${\displaystyle \sum _j{F}_{ji}}$,

$${\displaystyle \frac{d{q}_i}{dx}}={\displaystyle \frac{d(s_{i}T)}{dx}}=({\mathrm{s}}_{oi}-R\ln c_i+C_{\mathrm{vi}}\ln T){\displaystyle \frac{dT}{dx}}\hspace{0.17em}+T({\displaystyle \frac{d{s}_{0i}}{dx}}-{\displaystyle \frac{R}{c_i}}{\displaystyle \frac{d{c}_i}{dx}}+{\displaystyle \frac{C_{Vi}}{T}}{\displaystyle \frac{dT}{dx}})$$

After simplification, it is reduced to

$$d{s}_i={\displaystyle \frac{d{q}_i}{T}}-({\mathrm{s}}_{oi}-R\ln c_i+C_{\mathrm{vi}}\ln T){\displaystyle \frac{dT}{T}}$$

In addition, when heat is used to move a piston by a gas without equilibrium, the gas temperature decreases. This explains well-known Clausius inequality $d{s}_i\ge d{q}_i/T$. In the opposite situation of non-isolated systems driven by an external forces piston, it is possible that the entropy decreases because of different signs of external and entropic forces.

Without external forces, the flux after substitution becomes

$$J_i=U_i{c}_i{\displaystyle \frac{d{q}_i}{dx}}=U_i{c}_i[T({\displaystyle \frac{d{s}_{0i}}{dx}}-{\displaystyle \frac{R}{c_i}}{\displaystyle \frac{d{c}_i}{dx}})+({\mathrm{s}}_{oi}-R\ln c_i+C_{\mathrm{vi}}\ln T+C_{\mathrm{v}i}){\displaystyle \frac{dT}{dx}}]$$

(3)

Some simplifications are interesting:

In the homogeneous and isothermal system $d{s}_{0i}/dx=0$ and $dT\hspace{0.17em}/dx=0$, leading to Fick’s law of diffusion $J_i=-RT{U}_i{\displaystyle \frac{d{c}_i}{dx}}=-D_i{\displaystyle \frac{d{c}_i}{dx}}\hspace{0.17em}\hspace{0.17em}$ and the Fokker–Einstein relation $D_i=RT{U}_i$. Mass conservation law ${\displaystyle \frac{dc}{dt}}=-{\displaystyle \frac{dJ}{dx}}$ after substitution in the presence of fields leads to the Fokker–Planck–Smoluchowski equation ${\displaystyle \frac{\partial c}{dt}}=U{\displaystyle \frac{\partial }{\partial x}}[RT{\displaystyle \frac{\partial }{\partial x}}-{\displaystyle \sum _j{F}_j}]c$ [9]. Without external fields, it is reduced to the Second Fick’s law of diffusion

$${\displaystyle \frac{\partial c}{\partial t}}=RTU{\displaystyle \frac{{\partial }^{2}c}{\partial x^2}}$$

When diffusion-driven ion flux is balanced by an electric field-driven flux in the opposite direction, in equilibrium $J_{Fick}=-J_{Faraday}$ and $-RT{U}_i{\displaystyle \frac{d{c}_i}{dx}}=U_i{c}_i{z}_{i}F{\displaystyle \frac{d\psi }{dx}}$. After integration it gives the Nernst law: $\Delta \psi ={\displaystyle \frac{RT}{z_{i}F}}\ln {\displaystyle \frac{c_1}{c_2}}$. Similarly, for pressure and molar volume $\mathrm{v} ¯$ we have $\Delta P={\displaystyle \frac{RT}{ \mathrm{v} ¯}}\ln {\displaystyle \frac{c_1}{c_2}}\simeq {\displaystyle \frac{RT}{ \mathrm{v} ¯}}\Delta c$. Thus, for small $\Delta c$ we have the Van’t Hoff law for osmotic pressure [2]. Instead of logarithmic dependences for concentrations, for the balance of electric field- and pressure-driven fluxes we have ${\displaystyle \frac{d\psi }{dP}}=-{\displaystyle \frac{ \mathrm{v} ¯}{z_{i}F}}=const$. Similar types of equilibrium relations should be valid for other potential-based forces at constant temperature.

3.3. Thermodiffusion: Soret and Dufour Effects

If the initial concentration is constant, and we have as a driving factor only a temperature gradient along the space coordinate x, the process is called thermodiffusion, or the Soret effect. The flux should be

$$J_i=U_i{c}_i(C_{\mathrm{v}i}+s_{oi}-R\ln c_i+C_{\mathrm{v}i}\ln T){\displaystyle \frac{dT}{dx}}=-D_{Ti}{c}_i{\displaystyle \frac{dT}{dx}}$$

(4)

with thermodiffusion coefficient $D_{Ti}$. The Soret coefficient is the ratio $S_T=D_{Ti}/D$. The signs in front of terms with lnc and lnT are different, and we have two possible situations:

$a.\hspace{0.17em}\hspace{0.17em}\mathrm{If} \left|\ln c_i\right|\hspace{0.17em}<{\displaystyle \frac{s_{oi}+C_{\mathrm{v}i}(1+\ln T)}{R}}\hspace{0.17em}\hspace{0.17em}$, the temperature-driven flux is positive and directed to the hot side.

$\hspace{0.17em}b.\hspace{0.17em}\hspace{0.17em}\mathrm{If}\ln c_i>{\displaystyle \frac{s_{oi}+C_{\mathrm{v}i}(1+\ln T)}{R}}$, the flux is negative and directed to the cold side.

For ion transport through polymer membranes, the effect depends on both polymer and ion. For example, with 1 mM KCl solution at temperatures below 311 K the direction of flux through the phenolsulfonic acid membrane was from the hot to the cold side, and at higher temperatures the flux was from the cold to the hot side [10]. It is possible to estimate at what concentration the flux should change its direction. Assuming that $J_i=$0, $s_{oi}=0$, and T = 300 K, and using that for one degree of freedom $C_{\mathrm{v}i}=R/2$, we get c = 28.5 mol/m³ or 28.5 mM. Thus, our physicochemical mechanics approach leads to simple explanation why the flux changes its direction as the result of minor changes of temperature and concentration.

With time, a concentration gradient will be formed. In equilibrium of two driving factors, assuming for simplicity that ${\displaystyle \frac{d{s}_{0i}}{dx}}=0$, $C_{\mathrm{v}i}\ln T=0$, and $R\ln c_i=0$,

$$J_i=U_i{c}_i[T(-{\displaystyle \frac{R}{c_i}}{\displaystyle \frac{d{c}_i}{dx}})+(s_{0i}+C_{\mathrm{v}i}+C_{\mathrm{v}i}\ln T){\displaystyle \frac{dT}{dx}}]=0$$

$$(s_{0i}+C_{\mathrm{v}i})d\ln T=Rd\ln c_i$$

Further, using that established temperature difference is also small, after simplifications

$${\displaystyle \frac{\Delta T}{T_1}}\simeq {\displaystyle \frac{R}{s_{0i}+C_{\mathrm{v}i}}}{\displaystyle \frac{\Delta c_i}{c_{i1}}}$$

(5)

Usually, ${\displaystyle \frac{s_{0i}+C_{\mathrm{v}i}}{R}}>1$, and if $c_{i2}>c_{i1}$, we have $T_2>T_1$. Thus, the concentration gradient leads to small temperature gradient (known as the Dufour effect), and in considered conditions the higher temperature is in the area with higher concentration. The situation is different if $R\ln c_i$ is large. In this case, using Equation (3) in equilibrium and neglecting $s_{oi}$ and two terms with $C_{\mathrm{vi}}$

$$R{\displaystyle \frac{d\ln c_i}{dx}}=-R\ln c_i{\displaystyle \frac{d\ln T}{dx}}.$$

After integration, $\ln {\displaystyle \frac{T_1}{T_2}}=\ln {\displaystyle \frac{\ln c_{i2}}{\ln c_{i1}}}$ or ${\displaystyle \frac{T_1}{T_2}}={\displaystyle \frac{\ln c_{i2}}{\ln c_{i1}}}$. Now, if $c_{i2}>c_{i1}$, T₂ is less than T₁. Thus, if the concentration is large, and the process is driven by diffusion, we have a situation well-known in molecular physics when a substance is accumulated in a colder area. Evidently, there should be conditions (concentration and temperature) when the process changes its direction.

3.4. Thermoelectric Peltier–Seebeck and Thomson Effects

Further, it is easy to add electric forces and describe different thermoelectric effects, including Peltier–Seebeck and Thomson effects. For metals, electron concentration is high. Because of that, concentration gradient is low and the Peltier effect for electrons (the elementary charge $z=-1$) is described by

$$J_T=U_i{c}_i(C_{\mathrm{v}i}+s_{oi}-R\ln c_i+C_{\mathrm{v}i}\ln T){\displaystyle \frac{dT}{dx}}=-J_{\psi }=-U_i{c}_{i}F{\displaystyle \frac{d\psi }{dx}}$$

Finally,

$$\Delta T=-{\displaystyle \frac{F}{C_{\mathrm{v}i}+s_{oi}-R\ln c_i+C_{\mathrm{v}i}\ln T}}\Delta \psi$$

Temperatures should be higher in the point with lower voltage. Nevertheless, as it was with thermodiffusion, changes of material properties may increase role of entropic force and concentration gradient and may even change the sign of temperature dependence on voltage.

In turn, the Seebeck effect is the voltage, generated by temperature difference. The voltage is proportional to the temperature difference between the two junctions. The proportionality constant is known as the Seebeck coefficient. By convention, its sign is the sign of the potential of the cold end with respect to the hot end. Seeback coefficient is not a constant and depends on temperature. The temperature dependence of a commercial thermocouple is usually expressed as an empirical polynomial function in powers of temperature. Now we have a unified approach to describe both thermodiffusion and thermoelectric effects.

Before the equilibrium is reached, the heat absorbed or created is proportional to the electrical current. The proportionality constant is known as the Peltier coefficient. In the Thomson (Lord Kelvin) effect, heat is absorbed or produced when current flows in a material with a temperature gradient. Similar to chemical or electron flux, the heat should be proportional to both the electric current and the temperature gradient. Not surprisingly, the proportionality constant, known as the Thomson coefficient, is related to the Seebeck coefficient. We discussed here the simplest situation, but modern thermoelectric energy converters are based on semiconducting materials where voltage is generated at the contact area, and heat conductivity may be influenced by mobility of electrons, ions, and molecules, multiplied by their concentrations and molar heats carried by each of these components.

4. Discussion

It is well-known that Hamiltonian mechanics are the total of kinetic and potential energy of the system, while Lagrangian is their difference. Short comparison of Newtonian, Lagrangian, and Hamiltonian mechanics may be found in [5], Section 2.1–2.3. Instead of total kinetic energy of directed movements in mechanics, we are using the total of the heats carried by each component i and mainly determined by Brownian motion. For those influenced by external fields systems, we suggest a new Lagrangian $L_s={\displaystyle \sum T{s}_i-{\displaystyle \sum {\mathrm{v}}_i}}$, which is the difference between this total heat $TS={\displaystyle \sum T{s}_i}$ and total potential energy $V={\displaystyle \sum {\mathrm{v}}_i}$. As a result of any transport processes, total potential energy (taken with minus sign) decreases, dissipating into heat. The heat increases, and $L_s$ always increases, leading to

$${\displaystyle \frac{d{L}_s}{dt}}={\displaystyle \frac{d({\displaystyle \sum T{s}_i-{\displaystyle \sum {\mathrm{v}}_i})}}{dt}}\ge 0$$

(6)

$d{L}_s/dt$ is zero when potential energy balances heat, reminding the equipartition theorem in statistical physics. For constant temperature and without external fields, this more general formulation again leads to entropy increasing with time, i.e., it is valid for isolated systems under the Second Law of Thermodynamics. If so, the flux of each chemical component is proportional to the difference of its molar heat-based and potential forces, which are derivatives over the coordinate x:

$$J_i=U_i{c}_i{\displaystyle \frac{d}{dx}}(T{s}_i{-\mathrm{v}}_i)$$

(7)

If we divide both sides of this equation by concentration, we get an equation for velocity, which is an analog to an equation of motion in mechanics. To deal with additional degrees of freedom instead of vectors like mechanical forces one should deal with tensors. Note that if the First Law of Thermodynamics talks about energy preservation, and the Second Law about entropy increase, now we have a united law, which describes flux as a function of both entropy and potential energy gradients.

It is interesting to compare our approach with non-equilibrium thermodynamics [11] and more recently developed extended irreversible thermodynamics [12]. Entropy as a driving factor in phase transitions and its relation to reciprocal relations are discussed in [13]. The different signs in (7) remind minus in Onsager–Casimir reciprocal relations for magnetic field-driven transport [14,15]. The explanation is that both the magnetic forces and molar heat are related to velocity. Nevertheless, physicochemical mechanics and Onsager’s linear nonequilibrium thermodynamics are different. Traditional linear thermodynamics assumes that the flux is described by $J_k={\displaystyle \sum _j{L}_{ik}}{F}_j$. $F_j$ here are not molar physical forces (newton/mol), but so-called thermodynamic forces. To find these thermodynamic forces, one must use the rate of entropy production, $dS/dt={\displaystyle \sum _k{J}_k{F}_k={\displaystyle \sum _{j,k}{L}_{k,j}{F}_j{F}_k}}$. For example, for energy transport, the thermodynamic force has units of 1/T. For mass transport, the definition of thermodynamic force is not unique, and it may be $-\nabla {\displaystyle \frac{\mu }{T}}$ or $-{\displaystyle \frac{\nabla \mu }{T}}$. Linear dependence for flux is possible because the system is near equilibrium where the path may be approximated as a straight line. In physicochemical mechanics, all molar forces have the same units (newton/mol) and may be added. In equilibrium, two forces balance each other, which leads to the well-known equilibrium laws [5]. This cannot be done for thermodynamic forces, and Onsager’s nonequilibrium thermodynamics is not reduced to equilibrium thermodynamics. Transport coefficients $L_{kj}$ in general are not known, and all we know is that, for correctly found thermodynamic forces, $L_{kj}=L_{jk}$ [14,15].

Physicochemical mechanics easily leads to these relations, but it also shows that they are not valid for multicomponent diffusion because mobilities are different for different independent components [5]. If all physical forces are known, and the purpose is to find the rate of chemical transport, it is not necessary to calculate the rate of entropy production anymore. We also do not need, for example, Helmholtz’s free energy with its temperature and volume as natural variables.

Chemical reactions are different, and we need mobility along with an additional chemical reaction coordinate [16]. Usually, they are conducted at independent and constant temperature and pressure as variables, which leads to the Gibbs–Duhem equation, ${\displaystyle \sum _i{N}_{i}d{\mu }_i=}-SdT+VdP$. For chemical transport, both temperature and pressure may be driving factors, and now ${\displaystyle \sum _i{\mu }_{i}d{N}_i=}-TdS+PdV$ [5]. For steam engines, maximum Carnot’s efficiency near equilibrium is $(T_1-T_2)/T_1$, but importantly for biology voltage-driven, transmembrane ion transport does not need changes of temperature. Because of the balance of electric and friction forces at a steady state and constant temperature, this process has a thermodynamic efficiency of 50% [5].

Mechanical models and entropy of black holes have attracted the attention of theoreticians for more than 50 years [17,18]. Physicochemical mechanics gives an interesting prediction for black holes as non-isolated systems with gravitational forces. In strong gravitational fields, not only a coordinate, but also gravitational acceleration, g, may increase towards the inner part of the hole. If x also increases towards the inner area of the hole, $dg/dx>0$. In this case, g-based potential increases inside the hole, and the g-based forces and mass transport are also directed outside. The total flux is determined by the balance of two driving factors, $J_i=U_i{c}_i[{\displaystyle \frac{d(s_{i}T)}{dx}}-{\displaystyle \frac{d(M_{i}gh)}{dx}}]$. For heat-driven processes we need two partial derivatives. Similarly, for gravity we also need two partial derivatives. One is ${({\displaystyle \frac{\partial g}{\partial x}})}_{h=const}$, and another one is ${({\displaystyle \frac{\partial h}{\partial x}})}_{g=const}=1$. If the temperature is small and positive, the final flux will be determined by the balance of three driving forces, due to g, h, and $s_i$. The steady state is possible if for some reason these three forces balance each other. In the situation, when the entropy decreases inside the black hole due to gravity-induced ordering, this may lead to flux towards entropy increase and black hole evaporation with time.

We do not know much about black holes, but for the Earth atmosphere we must add terms related to the light of the Sun. Absorbed by the Earth, surface solar radiation leads to atmosphere heating from one side. In addition, light absorption by air molecules leads to photoreactions, and we can expect that transport of ion-radicals will be influenced by electromagnetic fields, as mentioned above in the thermoelectric effects. The entire process may be called gravitothermoelectrodiffusion.

Thus, the number of terms is increased, but the general principle is still the same: we should add all potential energy- and entropy-related terms with proper signs. After that, the total derivative along the space coordinate will give the total acting force, and the final steady state transport velocity or flux in the real media with friction will be proportional to this force. Note that we did not discuss the convection here. Instead of molecular transport, it is based on movement of macroscopic volumes gas or liquid, and it may be a dominant transport process because friction is important in this case only at the surface of this volume.

A controversial description of thermoelectric effects may be found in [19,20]. It assumed that it is an electrical charge and not an ion with this charge carries the electrical energy. Similarly, instead of molecules, it is entropy that carries thermal energy. The author also believes that Peltier effect is an electrically induced entropy current, though this effect was discovered in 1834, i.e., when even the word entropy did not exist yet. Further, he suggests using entropy capacity instead of heat capacity, though it remains unclear how to measure it. Moreover, it is believed that entropy is a storable thermal quantity, though as we know, it increases in spontaneous processes in isolated systems [20].

A quite different approach is under development in quantum mechanics. For example, entropy and information flow were discussed for quantum systems strongly coupled to baths [21]. Usually, these systems include electrons with spin and their entanglement, but it is not clear what the space coordinate, local temperature, and its gradient are in this case. This makes derivation of classical chemical transport laws describing diffusion, thermodiffusion, comparison with Onsager’s theory in nonequilibrium thermodynamics, and others at least not easy if not impossible. It is not surprising that Einstein argued that the accepted formulation of quantum mechanics must be incomplete.

5. Conclusions

Einstein also modestly wrote about the role of his relativity theory in physics: “It has not appreciably altered the predictions of theory, but it has considerably simplified the theoretical structure, i.e., derivation of laws, and—what is incomparably more important—it has considerably reduced the number of independent hypotheses forming the basis of theory” [22]. Our purpose was to show that physicochemical mechanics, with its reduced number of hypotheses and one transport equation, unifying both the First and the Second Law of Thermodynamics, does something similar for driven by entropy, temperature, and external forces linear transport and its equilibria. Previously we also described exponential kinetics of elementary reactions [16]. Based on ideas from classical mechanics together with motion of spin it is also possible to derive Schrödinger’s equation [23].